This randomized phase I trial studies the side effects and the best dose of carfilzomib when given together with or without romidepsin in treating patients with stage IA-IVB cutaneous T-cell lymphoma. Carfilzomib and romidepsin may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. It is not yet known whether giving carfilzomib alone is more effective than when given together with romidepsin.